# Reconstructions of the dark-energy equation of state and the inflationary potential

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## Abstract

We use a mathematical approach based on the constraints systems in order to reconstruct the equation of state and the inflationary potential for the inflaton field from the observed spectral indices for the density perturbations \(n_{s}\) and the tensor to scalar ratio *r*. From the astronomical data, we can observe that the measured values of these two indices lie on a two-dimensional surface. We express these indices in terms of the Hubble slow-roll parameters and we assume that \(n_{s}-1=h\left( r\right) \). For the function \(h\left( r\right) \), we consider three cases, where \(h\left( r\right) \) is constant, linear and quadratic, respectively. From this, we derive second-order equations whose solutions provide us with the explicit forms for the expansion scale-factor, the scalar-field potential, and the effective equation of state for the scalar field. Finally, we show that for there exist mappings which transform one cosmological solution to another and allow new solutions to be generated from existing ones.

## Keywords

Cosmology Scalar field Inflation## 1 Introduction

An ‘inflaton’ is a scalar field that can drive a period of acceleration in the early universe. Such a finite period of inflation [1, 2] can solve long-standing problems about the structure of the universe that would otherwise require special initial conditions [3, 4]. An inflaton provides a matter source that can display antigravitating behavior and so it could also be a candidate for the so-called the ‘dark energy’ that drives cosmological acceleration today. It is possible that these two eras of cosmological acceleration are connected, but so far there is no compelling theory about how that link might arise between two such widely separated energy scales.

Various inflationary self-interaction potentials for the inflaton have been proposed in the literature. Since they lead to different inflationary scenarios, particularly in respect of the density fluctuations produced, they have different observational consequences for the cosmic microwave background radiation, and this permits them to be finely constrained by observational data. Various inflaton potentials in general relativistic scalar field cosmology have been proposed in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] , while for inflationary models in other gravity theories, where there are more possibilities, see [1, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] and references therein.

The construction of the inflaton scalar field potential from observational data is an open problem of special interest. It provides critical information about the details of the allowed inflationary models and might provide clues as to the identity of the inflaton. In [31, 32, 33, 34, 35, 36, 37], the perturbative reconstruction approach was applied: the inflaton self-interaction potential, \(V\left( \phi \right) \), of the scalar field, \(\phi \), was reconstructed by considering a series expansion around a point \(\phi =\phi _{0}\), where the coefficients of the series expansion for the potentials are determined from the observable values of the scalar spectral index and the usual slow-roll parameters; for more details see [38]. Alternative approaches to the reconstruction of the scalar field potential include a stochastic perturbative approach in [39], or another perturbative approach in [40]. Two alternative methods for the reconstruction of the scalar field potential have been proposed in [41] and [42]. Specifically, in the latter work, an exponential of the scalar field’s Hubble function was considered and found to offer an efficient way to derive and constrain the power-spectrum observables [42]. By contrast, in [41], the scalar field potential was reconstructed for the Harrison–Zeldovich spectrum by solving the gravitational field equations along with the equation for the adiabatic scalar perturbations.

The slow-roll parameters and their relations to the spectral indices have been reconstructed in closed-form [43, 44, 45]. This is the approach that we will follow here to find the equation of state for the effective perfect fluid which corresponds to the scalar field with a self-interaction potential. While this approach is not so accurate as the previous approaches (because it depends on approximate relations between the spectral indices and the slow-roll parameters [38]) it can more easily reconstruct closed-form solutions for the inflationary potential and the expansion scale factor expansion. Furthermore, as we shall see in the first approximations for the models that we study, there exist mappings which transform the models to other equivalent models and their linearised fluctuations to the Harrison–Zeldovich spectrum. The plan of this paper is as follows.

In Sect. 2, we review scalar field cosmology in a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) universe and introduce the basic quantities and notations. In Sect. 3, we assume that the spectral index for the density perturbations, \(n_{s}\), and the tensor-to-scalar ratio, *r*, are related by a function such that \(n_{s}-1=h\left( r\right) \). For the defining function, \(h\left( r\right) ,\) we assume that it is either constant, linear or quadratic in *r*. Moreover, using the slow-roll expressions for these indices, we find ordinary differential equations whose solutions provide us with the inflationary scalar field potentials and the equation of state for the energy density and the pressure of the scalar field while the density perturbations to tensor-to-scalar ration diagrams are presented for the analytical solutions that we derive. Moreover, in Sect. 4 the values for the free parameters of the models are determined in order a late time attractor to exists such that the universe to escape from the inflation phase. Moreover, a transformation which relates the different models that we study is presented in Sect. 5. We show that our master equations are all maximally symmetric. This ensures that maps exist which can transform the solution of one inflationary model into another. This can be used to determine new inflationary solutions from known ones. A discussion of the results presented and our conclusions are given in the concluding Sect. 6.

## 2 Underlying equations and definitions

*t*. The comoving observers have \(u^{\mu }=\delta _{0}^{\mu }\) , so \(u^{\mu }u_{\mu }=-1\). The FLRW symmetries ensure \(\phi =\phi \left( t\right) \).

*q*, is given by the formula \(q=\frac{1}{2}\left( 1+3w_{\phi }\right) \) because, as the only matter source is the scalar field, we have \(w_{tot}=w_{\phi }\). The expansion of the universe is accelerated when \(q<0\), that is, \(w_{\phi }<-\frac{1}{3}\). Since \(V\left( \phi \right) >0\), a negative negative EoS parameter means that the potential dominates the kinetic term i.e., \(\frac{{\dot{\phi }}^{2}}{2}<V(\phi )\). Furthermore, in the limit \({\dot{\phi }}\rightarrow 0\) expression (8) gives \(w_{\phi } \rightarrow -1\), and the scalar field mimics the cosmological constant.

### 2.1 Analytical solution

*a*(

*t*) is the cosmic scale factor, can be solved for the scalar field, \(\phi \), and the potential, \(V\left( \phi \right) ,\) by the following formulae

^{1}

Lastly, using expression (19), all the slow-roll parameters can be expressed in terms of the parameter \(\varepsilon _{H}\) and its derivatives.

## 3 Reconstruction of the inflationary potential

From the recent data analysis by the Planck 2015 collaboration [4], the value of the spectral index for the density perturbations is \(n_{s}=0.968\pm 0.006,~\)while the range of the scalar spectral index is \(n_{s}^{\prime }=-0.003\pm 0.007\). The tensor to scalar ratio, *r*, has a value smaller than 0.11, i.e., \(r<0.11\).

*h*, then we define a differential equation, which can be used to construct the exact form for the FLRW spacetime (15), i.e., determine \(F\left( \omega \right) \), that satisfies the spectral index conditions. Hence, with the use of the solution presented in the previous section, the scalar field potential can also be derived.

*h*(

*r*), which include the three first terms of the last Taylor expansion for the function \(h\left( r\right) \). Hence, by substituting from (18) in (24) three master equations follow for each chosen form of \(h\left( r\right) \).

### 3.1 Constant index: \(n_{s}-1=-2n_{0}\)

Assume that the spectral index for the density perturbations is constant, with \(n_{s}-1=-2n_{0}\), where according to the Planck 2015 data at \(1\sigma \), \(n_{0}\) should be bounded in the range \(\,0.013\le n_{0}\le 0.019\). In the case where \(n_{0}=0\), i.e., \(n_{s}=1\), we have the Harrison–Zeldovich spectrum. These cases were studied before in [43, 44, 45].

#### 3.1.1 Zero \(n_{0}:~\)Harrison–Zeldovich spectrum

#### 3.1.2 Non-zero \(n_{0}\)

*A*,

*B*such that \(F_{0}=-\frac{A}{1+B}\) and \(B=1+\frac{2}{3}n_{0}\).

*t*. Moreover, for the potential (35), we have that for large values of \(\phi \), the potential becomes exponential, that is,

*r*are calculated below.

#### 3.1.3 Observational constraints

*r*depends on the number of e-folds. Furthermore, the integration constants are non-essential and fix the value of the scale factor at the end of the inflation. In Fig. 1 the \(n_{s}-r\) plot is presented for the expressions (39) and for \(n_{0}\in \left( 0,0.02\right) \), \(\ N_{e}\in \left[ 50,60\right] \). Note that for values of \(n_{0}\) where \(n_{s}\) is constrained by the Planck 2015 data, it follows that \(r<0.11\) for very large values of \(N_{e}\), while for the number of e-folds that we considered in the figure \(r>0.11\).

Furthermore, in the case of the Harrison–Zeldovich spectrum, that is, \(n_{0}=0\), we calculate \(r=\frac{10}{1-2N_{e}}\), hence \(r<0.11\) when \(N_{e}>50\). As before, the integration constants (now \(F_{1}\) and \(\omega _{0} \)) specify only the value of \(\omega _{f}\) at which inflation ends. We calculate that \(\omega _{f}=3+\omega _{0}\).

### 3.2 Linear expression: \(n_{s}-1\simeq r\)

We continue now by taking the more general ansatz, \(n_{s}-1=-2n_{1} \varepsilon _{H}-2n_{0}\); that is, the spectral index \(n_{s}\) depends linearly on the tensor to scalar ratio, *r*. Recall that \(r=10\varepsilon _{H}\); so in the limit in which \(n_{1}\rightarrow 0\) we are in a situation where \(n_{s}-1=~const\). We study two cases: \(n_{0}=0\) and \(n_{0}\ne 0\).

#### 3.2.1 Zero \(n_{0}\):

#### 3.2.2 Non-zero \(n_{0}\):

#### 3.2.3 Observational constraints

We continue our analysis with a more general case in which the relation between \(n_{s}\) and *r* is parabolic.

### 3.3 Parabolic: \(n_{s}-1\simeq r^{2}\)

#### 3.3.1 Observational constraints

*r*are given in terms of the number of e-folds by

## 4 Conditions to escape from Inflation

We proceed by considering the cases (a) \(n_{2}=0\) and (b) \(n_{2}\ne 0\), where the number of critical points differs.

### 4.1 Subcase \(n_{2}=0\)

Let as assume the simple case which corresponds to the master equation (45); that is, \(n_{2}=0~\) and we assume that \(n_{1}\ne 1\). In that consideration, the critical points of the system are the \(\varepsilon _{H}^{\left( 0\right) }~\) and \(\varepsilon _{H}^{\left( 1\right) }~\)of (74).

As far as concerns the stability of these points, we find that point \(\varepsilon _{H}^{\left( 1\right) }\) is the unique attractor of the equation when \(n_{0}>0\), and \(\varepsilon _{H}^{\left( 1\right) }\) describes a point without acceleration when \(n_{1}<1\) and \(n_{0}>1-n_{1}\). On the other hand, when \(n_{0}<0\), the unique attractor of the system is the de Sitter point \(\varepsilon _{H}^{\left( 0\right) },\) although in this case the model does not provide an exit from inflation.

### 4.2 Subcase \(n_{2}\ne 0\)

For \(n_{2}\ne 0,\) a necessary condition for an exit from the inflation to occur, is that the critical points \(\varepsilon _{H}^{\left( \pm \right) }\) are real; that is, \(4n_{0}n_{2}\ge -\frac{\left( 1-n_{1}\right) ^{2}}{4}\). In the special limit in which \(n_{0}=0\), the points \(\varepsilon _{H}^{\left( \pm \right) }\) reduce to \(\varepsilon _{H}^{\left( 0\right) }\) and \(\varepsilon _{H}^{\left( 2\right) }=\frac{n_{1}-1}{n_{2}}\). In that case, the two points are stable when \(n_{2}>0\), and \(\varepsilon _{H}^{\left( 2\right) }\) is positive for any value of \(n_{1}>1\).

## 5 Equivalent transformations

It is interesting that, when we set \(n_{s}-1=0\), the scalar field mimics the generalized Chaplygin gas (28) with \(\lambda =2\). Yet, when we assumed that \(\lambda \ne 2\) in the equation of state of the generalized Chaplygin gas, we found that \(n_{s}-1=-2n_{1}\varepsilon _{H}\), where \(\lambda =2-n_{1}\). These two models are the solutions of the two different master equations, (26) and (40), respectively. Yet, these two equations are different for \(n_{1}\ne 0\), we observe that there exists a transformation \(F\left( \omega \right) \rightarrow {\bar{F}}\left( \omega \right) \), allowing eq. (26) to be written in the form of (40) and vice versa.

On the other hand, it is important to mention that eq. (40) can be written in the form of (26) under the simple change of variable \(\omega =\frac{3}{n_{0}}\ln \left( {\bar{\omega }}\right) \). The same transformation can be applied in the master equation, (45), which is transformed into equation (40). Moreover, if we also apply the transformation (78) to (45), then the latter takes the form of the master equation, (26).

The existence of transformations of this kind, which transform the one model into another, is not a coincidence. The master equations (26), (31), (40) and (45) are maximally symmetric. In particular they are invariant under the action of one-parameter point transformations (Lie point symmetries) which form the \(SL\left( 3,R\right) \) Lie algebra.^{2}

Consider now the classical Newtonian analogue of a free particle and an observer whose measuring instruments for time and distance are not linear. By using the measured data of the observer we reach in the conclusion that it is not a free particle. On the other hand, in the classical system of the harmonic oscillator an observer with nonlinear measuring instruments can conclude that the system observed is that of a free particle, or that of the damped oscillator or another system. From the different observations, various models can be constructed. However, all these different models describe the same classical system and the master equations are invariant under the same group of point transformations but in different parametrization.

In the master equations that we studied there is neither position nor time variables: the independent variable is the scale factor \(\omega =6\ln a\), and the Hubble function is the dependent variable, \(H\left( a\right) \). Therefore, we can say that at the level of the first-order approximation for the spectral indices, various representations of the variables \(\left\{ a,H\left( a\right) \right\} \) provide different observable values for the spectral indices. This property is violated when we consider the second-order approximation.

Transformations of this kind are well-known in physics. For instance, the Darboux transformation for the Schrödinger equation [60] is just a point transformation that relates linear equations with maximal symmetry; that is, it belongs to exactly the same category of transformations that we discuss here. A special characteristic of the Darboux transformation is that it preserves the form of the equation but the potential in the Schrödinger equation changes. An application of the Darboux transformation for the determination of exactly solvable cosmological models can be found in [61].

Transformations which keep the form of our master equation exist. We do not have potential terms in the master equations but there are transformations which change the constant coefficients appearing there while retaining the form of the master equations.

Furthermore, for the more general case that we studied (the master equation of eq. (60)) it is easy to see that for \(n_{2}n_{0}\ne 0\), eq. (60) admits eight Lie point symmetries; that is, it is maximally symmetric. Hence, there exists a mapping \(\left\{ \omega ,F\left( \omega \right) \right\} \rightarrow \left\{ \Omega ,\Phi \left( \Omega \right) \right\} \) which transforms the master equation (60) to that of a free particle, or to any other maximally symmetric equation—such as the other master equations we studied above. Of course, this result can be used to derive closed-form solutions in other models with a maximally symmetric master equation.

Recall that a map in the space of the variables which transforms one solution to any other solution was also found in [57]. However, while both maps transform solutions into solutions, the one that we have discussed here, transforms not only solutions into solutions but systems of dynamical equations into equivalent systems^{3}. In order to reflect that latter property, the map is called an equivalent point transformation.

The elements of the \(SL\left( 3,R\right) \)—except for the transformations which relate algebraic equivalent equations— provide us with important physical information about the system under study. One of these properties which arises from eq. (26) is the well-known scale invariance of the Harrison–Zeldovich spectrum, regarding which it can easily be seen that equation (26) is invariant under transformations \(\omega =\omega ^{\prime }+\omega _{0}\) or \(\omega =\omega ^{\prime }e^{{\bar{\omega }}_{0}}\), where these two transformations are related with the symmetry vectors \(\partial _{\omega }\) and \(\omega \partial _{\omega }\). In particular, every element of the \(SL\left( 3,R\right) \) is related to a point transformation which leaves the differential equation, and consequently the solution, invariant. Moreover, with a different reparameterization of the \(SL\left( 3,R\right) \), for equivalent models, the physical interpretation of the invariant point transformations can change between the different models.

## 6 Conclusions

In scalar-field cosmology, the dark-energy EoS and the inflationary scalar-field potential have been reconstructed from the spectral index, \(n_{s}\). From the Planck 2015 data analysis, it is known that the observable variables—the tensor-to-scalar ratio, *r*, and the spectral index for the density perturbations, \(n_{s}\)—form a surface in the \(n_{s}-r\) plane. Furthermore, these two observable variables can be expressed in terms of the slow-roll parameters and their derivatives. Therefore, the ansatz that the spectral index for the density perturbations is related with the tensor-to-scalar ratio, \(\left( n_{s}-1\right) =h\left( r\right) \), provides a differential (master) equation whose solution defines the corresponding cosmological model.

In this paper, we assumed \(n_{s}\) to be given in the first approximation by a function \(h\left( r\right) \) that it is: (a) constant, (b) linear, and (c) quadratic, respectively. In order for the first-order approximation to be valid the free parameters which have been introduced by the function \(h\left( r\right) \) have to satisfy some consistency conditions.

We work with the HSR parameters. The case in which \(h\left( r\right) \) is constant, that is, \(n_{s}-1=-2n_{0},\,\) is one that has been studied before in the literature and, in the limit, \(n_{0}=0\), corresponds to the Harrison–Zeldovich spectrum. The differential equation which follows provides the scalar factor to be that of a specific intermediate inflation, \(a\left( t\right) \simeq \exp \left( a_{1}t^{2/3}\right) \), while the corresponding perfect fluid satisfies the equation of state (28). On the other hand, for nonzero \(n_{0}\), we found that the scalar field satisfies an EoS given by expression (48) for \(\lambda =2,\) which includes expression (28). For the scalar-field potential, the construction looks similar, and for \(n_{0}=0\) the potential is given in terms of polynomials of the field \(\phi \), and for \(n_{0}\ne 0\) in terms of hyperbolic trigonometric functions.

As a second generalization, we assumed \(h\left( r\right) \) to be the linear function, \(h\left( r\right) =-\frac{n_{1}}{5}r-2n_{0}\). Now, the models derived from the differential equation \(n-1=h\left( r\right) ,\) in the first-order approximation, are the generalized Chaplygin gases, (28) and (48), for \(n_{0}=0\) and \(n_{0}\ne 0,\) respectively; where now the power \(\lambda \) in the equations of state is related to the value of \(n_{1}\), by \(\lambda =2-n_{1}\).

Finally, the case in which \(h\left( r\right) \) is a quadratic polynomial was considered and two new equations of state which generalize the Chaplygin gas were derived. Exact examples displaying a generalised sudden singularity of the type identified by Barrow and Graham [62] for inflationary scalar fields with fractional potentials were found here. Lastly, the ranges for the values of the free parameters of the models have been considered which permit the universe to escape from the inflationary phase.

It is important to mention that in this work we have assumed that we are in the inflationary epoch and so the equation of state parameters, or equivalently the scalar field potentials that we reconstructed, can be seen as the leading order terms, or attractors, of a more general equation of state parameter which describes the whole evolution of the universe.

It is particularly interesting that the master equations we derived in our study are second-order differential equations of maximal symmetry. Hence, they are invariant under the action of point transformations with generators given by the elements of the \(SL\left( 3,R\right) \) algebra. Every master equation defines a representation of the \(SL\left( 3,R\right) \) algebra and the map which changes the representation transforms the master equation to the corresponding master equation of another model. This relates explicitly the form of the line elements for the various cosmological models. The transformation which performs the change is a projective transformation in the jet-space of the master equation; that is, a map in the space of the dependent variable \(F\left( \omega \right) \) and the spacetime variable \(\omega \) – we recall that \(dt=e^{-F\left( \omega \right) /2}d\omega \) and \(a\left( t\right) =e^{\omega /6}\).

In a forthcoming work we will investigate whether the latter result can be extended to the case in which the master equation, \(n_{s}-1=h\left( r\right) \), is defined by higher-order approximations for the spectral indices.

## Footnotes

- 1.
Where a prime “\(^{\prime }\)” denotes the total derivative with respect to \(\omega \).

- 2.
According to Lie’s Theorem, any second-order equation which admits the elements of the \(SL\left( 3,R\right) \) algebra as symmetries is equivalent to the equation of a free particle and all the maximally symmetric equations commute [58]. The map is the one which transforms the admitted \(SL\left( 3,R\right) \) Lie algebra among the different representations of the admitted equations, for more details see [59].

- 3.
Other transformations which belong to these families of transformations are presented in [18].

## Notes

### Acknowledgements

JDB is supported by the Science and Technology Facilities Council of the United Kingdom (STFC). AP acknowledges the financial support of FONDECYT Grant No. 3160121. AP thanks the Durban University of Technology for the hospitality provided while part of this work was performed.

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